Atlas of Fundamental Concepts

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Representable presheaf

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Summary

A presheaf naturally isomorphic to Hom⁑C(βˆ’,c)\operatorname{Hom}_{\mathcal C}(-,c) for an object cc of a locally small category C\mathcal C.

Record metadata

Canonical name
Representable presheaf
Concept ID
representable_presheaf
Content version
2.3.0
Primary field
Mathematics
Fields
Mathematics
Primary domain
Category theory
Domains
Category theory
Alternate terminology
No alternate terminology is currently authored for this record.
Concept type
Functorial representation

Concept sources

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

PresheafRepresentable presheaf

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID
category_presheaf_to_representable_presheaf
Relation type
Classification classification
Direction
source β†’ target
Endpoint roles
source: Has subtype; target: Classified as
Authored annotation
represented by an object

Authored explanation

Representable presheaves are the subclass of presheaves whose values and restriction maps are naturally isomorphic to hom-sets into one fixed representing object.

How to interpret this relation type

The target is a member or subtype of the broader source class.

Relation sources

Small categoryRepresentable presheaf

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID
category_small_category_to_representable_presheaf
Relation type
Canonical construction canonical-construction
Direction
source β†’ target
Endpoint roles
source: Canonically constructs; target: Canonically constructed from
Authored annotation
form a hom presheaf

Authored explanation

Fixing an object cc of a small category canonically gives the contravariant hom-functor Hom⁑C(βˆ’,c):Copβ†’Set\operatorname{Hom}_{\mathcal C}(-,c):\mathcal C^{\mathrm{op}}\to\mathbf{Set}.

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

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